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Repeated measures design

repeated measures design, repeated measures design example
Repeated measures design uses the same subjects with every branch of research, including the control[1] For instance, repeated measurements are collected in a longitudinal study in which change over time is assessed Other non-repeated measures studies compare the same measure under two or more different conditions For instance, to test the effects of caffeine on cognitive function, a subject's math ability might be tested once after they consume caffeine and another time when they consume a placebo


  • 1 Crossover studies
  • 2 Uses
  • 3 Order effects
  • 4 Counterbalancing
  • 5 Limitations
  • 6 Repeated measures ANOVA
    • 61 Partitioning of error
    • 62 Assumptions
    • 63 F test
    • 64 Effect size
    • 65 Cautions
  • 7 See also
  • 8 Notes
  • 9 References
    • 91 Design and analysis of experiments
    • 92 Exploration of longitudinal data
  • 10 External links

Crossover studies

Main article: Crossover study

A popular repeated-measures is the crossover study A crossover study is a longitudinal study in which subjects receive a sequence of different treatments or exposures While crossover studies can be observational studies, many important crossover studies are controlled experiments Crossover designs are common for experiments in many scientific disciplines, for example psychology, education, pharmaceutical science, and health-care, especially medicine

Randomized, controlled, crossover experiments are especially important in health care In a randomized clinical trial, the subjects are randomly assigned treatments When such a trial is a repeated measures design, the subjects are randomly assigned to a sequence of treatments A crossover clinical trial is a repeated-measures design in which each patient is randomly assigned to a sequence of treatments, including at least two treatments of which one may be a standard treatment or a placebo: Thus each patient crosses over from one treatment to another

Nearly all crossover designs have "balance", which means that all subjects should receive the same number of treatments and that all subjects participate for the same number of periods In most crossover trials, each subject receives all treatments

However, many repeated-measures designs are not crossovers: the longitudinal study of the sequential effects of repeated treatments need not use any "crossover", for example Vonesh & Chinchilli; Jones & Kenward


  • Limited number of participants —The repeated measure design reduces the variance of estimates of treatment-effects, allowing statistical inference to be made with fewer subjects[2]
  • Efficiency—Repeated measure designs allow many experiments to be completed more quickly, as fewer groups need to be trained to complete an entire experiment For example, experiments in which each condition takes only a few minutes, whereas the training to complete the tasks take as much, if not more time
  • Longitudinal analysis—Repeated measure designs allow researchers to monitor how participants change over time, both long- and short-term situations

Order effects

Order effects may occur when a participant in an experiment is able to perform a task and then perform it again Examples of order effects include performance improvement or decline in performance, which may be due to learning effects, boredom or fatigue The impact of order effects may be smaller in long-term longitudinal studies or by counterbalancing using a crossover design


In this technique, two groups each perform the same tasks or experience the same conditions, but in reverse order With two tasks or conditions, four groups are formed

Counter Balancing
Task/Condition Task/Condition Remarks
Group A 1 2 Group A performs Task/Condition 1 first, then Task/Condition 2
Group B 2 1 Group B performs Task/Condition 2 first, then Task/Condition 1

Counterbalancing attempts to take account of two important sources of systematic variation in this type of design: practice and boredom effects Both might otherwise lead to different performance of participants due to familiarity with or tiredness to the treatments


It may not be possible for each participant to be in all conditions of the experiment ie time constraints, location of experiment, etc Severely diseased subjects tend to drop out of longitudinal studies, potentially biasing the results In these cases mixed effects models would be preferable as they can deal with missing values

Mean regression may affect conditions with significant repetitions Maturation may affect studies that extend over time Events outside the experiment may change the response between repetitions

Repeated measures ANOVA

Repeated measures analysis of variance rANOVA is a commonly used statistical approach to repeated measure designs[3] With such designs, the repeated-measure factor the qualitative independent variable is the within-subjects factor, while the dependent quantitative variable on which each participant is measured is the dependent variable

Partitioning of error

One of the greatest advantages to rANOVA, as is the case with repeated measures designs in general, is the ability to partition out variability due to individual differences Consider the general structure of the F-statistic:

F = MSTreatment / MSError = SSTreatment/dfTreatment/SSError/dfError

In a between-subjects design there is an element of variance due to individual difference that is combined with the treatment and error terms:

SSTotal = SSTreatment + SSError

dfTotal = n-1

In a repeated measures design it is possible to partition subject variability from the treatment and error terms In such a case, variability can be broken down into between-treatments variability or within-subjects effects, excluding individual differences and within-treatments variability The within-treatments variability can be further partitioned into between-subjects variability individual differences and error excluding the individual differences:[4]

SSTotal = SSTreatment excluding individual difference + SSSubjects + SSError

dfTotal = dfTreatment within subjects + dfbetween subjects + dferror = k-1 + n-1 + n-kn-1

In reference to the general structure of the F-statistic, it is clear that by partitioning out the between-subjects variability, the F-value will increase because the sum of squares error term will be smaller resulting in a smaller MSError It is noteworthy that partitioning variability reduces degrees of freedom from the F-test, therefore the between-subjects variability must be significant enough to offset the loss in degrees of freedom If between-subjects variability is small this process may actually reduce the F-value[4]


As with all statistical analyses, specific assumptions should be met to justify the use of this test Violations can moderately to severely affect results and often lead to an inflation of type 1 error With the rANOVA, standard univariate and multivariate assumptions apply[5] The univariate assumptions are:

  • Normality—For each level of the within-subjects factor, the dependent variable must have a normal distribution
  • Sphericity—Difference scores computed between two levels of a within-subjects factor must have the same variance for the comparison of any two levels This assumption only applies if there are more than 2 levels of the independent variable
  • Randomness—Cases should be derived from a random sample, and scores from different participants should be independent of each other

The rANOVA also requires that certain multivariate assumptions be met, because a multivariate test is conducted on difference scores These assumptions include:

  • Multivariate normality—The difference scores are multivariately normally distributed in the population
  • Randomness—Individual cases should be derived from a random sample, and the difference scores for each participant are independent from those of another participant

F test

As with other analysis of variance tests, the rANOVA makes use of an F statistic to determine significance Depending on the number of within-subjects factors and assumption violations, it is necessary to select the most appropriate of three tests:[5]

  • Standard Univariate ANOVA F test—This test is commonly used given only two levels of the within-subjects factor ie time point 1 and time point 2 This test is not recommended given more than 2 levels of the within-subjects factor because the assumption of sphericity is commonly violated in such cases
  • Alternative Univariate test[6]—These tests account for violations to the assumption of sphericity, and can be used when the within-subjects factor exceeds 2 levels The F statistic is the same as in the Standard Univariate ANOVA F test, but is associated with a more accurate p-value This correction is done by adjusting the degrees of freedom downward for determining the critical F value Two corrections are commonly used—The Greenhouse-Geisser correction and the Huynh-Feldt correction The Greenhouse-Geisser correction is more conservative, but addresses a common issue of increasing variability over time in a repeated-measures design[7] The Huynh-Feldt correction is less conservative, but does not address issues of increasing variability It has been suggested that lower Huynh-Feldt be used with smaller departures from sphericity, while Greenhouse-Geisser be used when the departures are large
  • Multivariate Test—This test does not assume sphericity, but is also highly conservative

Effect size

One of the most commonly reported effect size statistics for rANOVA is partial eta-squared ηp2 It is also common to use the multivariate η2 when the assumption of sphericity has been violated, and the multivariate test statistic is reported A third effect size statistic that is reported is the generalized η2, which is comparable to ηp2 in a one-way repeated measures ANOVA It has been shown to be a better estimate of effect size with other within-subjects tests[8][9]


rANOVA is not always the best statistical analysis for repeated measure designs The rANOVA is vulnerable to effects from missing values, imputation, unequivalent time points between subjects and violations of sphericity[10] These issues can result in sampling bias and inflated rates of Type I error[11] In such cases it may be better to consider use of a linear mixed model[12]

See also

  • Analysis of variance
  • Clinical trial protocol
  • Crossover study
  • Design of experiments
  • Glossary of experimental design
  • Longitudinal study
  • Growth curve
  • Missing data
  • Mixed models
  • Multivariate analysis
  • Observational study
  • Optimal design
  • Panel analysis
  • Panel data
  • Panel study
  • Randomization
  • Randomized controlled trial
  • Repeated measures design
  • Sequence
  • Statistical inference
  • Treatment effect


  1. ^ Shuttleworth, Martyn 2009-11-26 "Repeated Measures Design" Experiment-resourcescom Retrieved 2013-09-02 
  2. ^ Barret, Julia R 2013 "Particulate Matter and Cardiovascular Disease: Researchers Turn an Eye toward Microvascular Changes" Environmental Health Perspectives 121: a282 doi:101289/ehp121-A282 PMC 3764084  PMID 24004855 
  3. ^ Gueorguieva; Krystal 2004 "Move Over ANOVA" Arch Gen Psychiatry 61: 310 doi:101001/archpsyc613310 
  4. ^ a b Howell, David C 2010 Statistical methods for psychology 7th ed Belmont, CA: Thomson Wadsworth ISBN 978-0-495-59784-1 
  5. ^ a b Salkind, Samuel B Green, Neil J Using SPSS for Windows and Macintosh : analyzing and understanding data 6th ed Boston: Prentice Hall ISBN 978-0-205-02040-9 
  6. ^ Vasey; Thayer 1987 "The Continuing Problem of False Positives in Repeated Measures ANOVA in Psychophysiology: A Multivariate Solution" Psychophysiology 24: 479–486 doi:101111/j1469-89861987tb00324x 
  7. ^ Park 1993 "A comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements" Stat Med 12: 1723–1732 doi:101002/sim4780121807 
  8. ^ Bakeman 2005 "Recommended effect size statistics for repeated measures designs" Behavior Research Methods 37 3: 379–384 doi:103758/bf03192707 
  9. ^ Olejnik; Algina 2003 "Generalized eta and omega squared statistics: Measures of effect size for some common research designs" Psychological Methods 8: 434–447 doi:101037/1082-989x84434 
  10. ^ Gueorguieva; Krystal 2004 "Move Over ANOVA" Arch Gen Psychiatry 61: 310–317 doi:101001/archpsyc613310 
  11. ^ Muller; Barton 1989 "Approximate Power for Repeated -Measures ANOVA lacking sphericity" Journal of the American Statistical Association 84 406: 549–555 doi:101080/01621459198910478802 
  12. ^ Kreuger; Tian 2004 "A comparison of the general linear mixed model and repeated measures ANOVA using a dataset with multiple missing data points" Biological Research for Nursing 6: 151–157 doi:101177/1099800404267682 


Design and analysis of experiments

  • Jones, Byron; Kenward, Michael G 2003 Design and Analysis of Cross-Over Trials Second ed London: Chapman and Hall 
  • Vonesh, Edward F & Chinchilli, Vernon G 1997 Linear and Nonlinear Models for the Analysis of Repeated Measurements London: Chapman and Hall 

Exploration of longitudinal data

  • Davidian, Marie; David M Giltinan 1995 Nonlinear Models for Repeated Measurement Data Chapman & Hall/CRC Monographs on Statistics & Applied Probability ISBN 978-0-412-98341-2 
  • Fitzmaurice, Garrett, Davidian, Marie, Verbeke, Geert and Molenberghs, Geert, eds 2008 Longitudinal Data Analysis Boca Raton, Florida: Chapman and Hall/CRC ISBN 1-58488-658-7 CS1 maint: Uses editors parameter link
  • Jones, Byron; Kenward, Michael G 2003 Design and Analysis of Cross-Over Trials Second ed London: Chapman and Hall 
  • Kim, Kevin & Timm, Neil 2007 ""Restricted MGLM and growth curve model" Chapter 7" Univariate and multivariate general linear models: Theory and applications with SAS with 1 CD-ROM for Windows and UNIX Statistics: Textbooks and Monographs Second ed Boca Raton, Florida: Chapman & Hall/CRC ISBN 978-1-58488-634-1 
  • Kollo, Tõnu & von Rosen, Dietrich 2005 ""Multivariate linear models" chapter 4, especially "The Growth curve model and extensions" Chapter 41" Advanced multivariate statistics with matrices Mathematics and its applications 579 New York: Springer ISBN 978-1-4020-3418-3 
  • Kshirsagar, Anant M & Smith, William Boyce 1995 Growth curves Statistics: Textbooks and Monographs 145 New York: Marcel Dekker, Inc ISBN 0-8247-9341-2 
  • Pan, Jian-Xin & Fang, Kai-Tai 2002 Growth curve models and statistical diagnostics Springer Series in Statistics New York: Springer-Verlag ISBN 0-387-95053-2 
  • Seber, G A F & Wild, C J 1989 ""Growth models Chapter 7"" Nonlinear regression Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics New York: John Wiley & Sons, Inc pp 325–367 ISBN 0-471-61760-1 
  • Timm, Neil H 2002 ""The general MANOVA model GMANOVA" Chapter 36d" Applied multivariate analysis Springer Texts in Statistics New York: Springer-Verlag ISBN 0-387-95347-7 
  • Vonesh, Edward F & Chinchilli, Vernon G 1997 Linear and Nonlinear Models for the Analysis of Repeated Measurements London: Chapman and Hall  Comprehensive treatment of theory and practice
  • Conaway, M 1999, October 11 Repeated Measures Design Retrieved February 18, 2008, from http://biostatmcvanderbiltedu/twiki/pub/Main/ClinStat/repmeasPDF
  • Minke, A 1997, January Conducting Repeated Measures Analyses: Experimental Design Considerations Retrieved February 18, 2008, from Ericaenet: http://ericaenet/ft/tamu/Rmhtm
  • Shaughnessy, J J 2006 Research Methods in Psychology New York: McGraw-Hill

External links

  • Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R University of Southampton

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